This paper is devoted to multistage games with pairwise interactions. The case of complete graph is considered, the vertices of which are players, and the edges are the connections between them. The characteristic function is introduced and its supermodularity is proved for single-stage game. A new approach for construction of characteristic function of multistage game is proposed, based on the use of values of characteristic functions of single-stage games. On the basis of the newly constructed characteristic function, the optimality principle is introduced, which is an analogue of the core, and its strong time-consistency proved. The work is illustrated by an example.